Article — Potential Energy Calculator
The potential energy calculator and stored energy in physics
Potential energy is the energy stored in an object's position or configuration. For gravity, PE = mgh: mass times gravitational acceleration times height. For springs, PE = ½kx²: half the spring constant times displacement squared. The SI unit is the joule (J). On Earth, raising a 1 kg mass by 1 meter stores 9.81 J of gravitational potential energy. Total mechanical energy KE + PE stays constant in the absence of friction, the basis of every roller coaster and hydroelectric plant.
The potential energy calculator covers the two everyday cases: gravitational and elastic. It also lets you switch planets, so you can see how much energy storage a hilltop on Mars provides compared with Earth.
What is potential energy?
Potential energy is what an object has because of where it is or how it is configured. It is the counterpart to kinetic energy, which depends on motion. A book on a shelf has potential energy. A stretched bowstring has potential energy. A compressed spring under your desk chair has potential energy. None of these objects move, yet each holds energy that can be released later.
The concept was developed in the 18th and 19th centuries by mathematicians and physicists working on mechanics — Daniel Bernoulli, Joseph-Louis Lagrange, and William Rankine among them. Rankine coined the term "potential energy" in the 1850s. The deeper insight, conservation of energy, was nailed down by Hermann von Helmholtz in 1847 and remains one of the most rigorously tested laws in physics.
A typical hydroelectric dam stores water hundreds of meters above its turbines. Hoover Dam holds 35 billion cubic meters of water at an average head of 175 m, storing roughly 6 × 10^16 joules of gravitational potential energy — equivalent to 14 million tons of TNT.
Gravitational potential energy explained
Gravitational potential energy comes from a mass's position in a gravitational field. The formula PE = mgh measures the energy relative to a chosen reference height. Pick any horizontal plane and call that h = 0. Heights above the plane give positive PE; heights below give negative PE. The reference choice is arbitrary because only changes in PE have physical meaning — what matters is how much PE converts to KE as the object moves.
PE = mgh on a planetm = PE / (gh) solve for massh = PE / (mg) solve for height1 J = 1 kg lifted ~0.1 m on Earth physical scaleThe Earth value g = 9.80665 m/s² comes from a 1901 international agreement. The actual local value varies slightly with latitude and altitude. For potential energy calculations to four significant figures, 9.81 is enough. The potential energy calculator includes a planet picker because gravity on Mars (3.71 m/s²) is just 38 percent of Earth's, while Jupiter's (24.79 m/s²) is 2.5 times Earth's.
Elastic potential energy and spring storage
An elastic object — a spring, a rubber band, a bent ruler — stores energy when deformed. Hooke's law says the restoring force is proportional to the displacement: F = kx, where k is the spring constant (N/m). The work done in stretching the spring by x is the integral of that force, ∫₀ˣ kx dx = ½kx². That energy is recovered when the spring snaps back.
The x-squared scaling means double the stretch stores four times the energy. This is why archery bows hit a sharp limit — pull a bow too far and it shatters. The same scaling explains why springs in car suspensions are sized for the heaviest expected loads: each extra centimeter of compression demands four times the energy as the centimeter before.
The potential energy formula in detail
Both formulas, PE = mgh and PE = ½kx², are derived by the same logic: integrate the relevant force over the displacement. For gravity the force is constant (mg for small heights), so the integral gives mgh. For springs the force is linear in displacement, so the integral picks up the factor of ½ and the square.
More general fields use a potential energy function U(x) whose negative gradient gives the force: F = -dU/dx. Gravitational PE near Earth uses U = mgh. Far from Earth, the inverse-square gravity law gives U = -Gm1m2/r. The potential energy calculator uses the near-surface approximation, which is accurate within a few percent for heights up to thousands of kilometers.
How to calculate potential energy step by step
Imagine lifting a 20 kg suitcase onto a 1 m shelf. PE = 20 × 9.81 × 1 = 196 J. Not much — less than a sip of orange juice. Now imagine raising the same suitcase to the top of the Empire State Building, 381 m. PE = 20 × 9.81 × 381 = 74,754 J = 75 kJ. The energy budget reflects the work required: every additional meter of height costs another 196 J.
For a spring problem, a car suspension might have a spring constant of 30 kN/m. Compressing it by 5 cm stores ½ × 30000 × 0.05² = 37.5 J. If both front springs compress simultaneously, the energy doubles to 75 J — soaked up and released over each road bump.
Conservation of energy lets you compute final velocity without using kinematics. If PE at the top of a slide equals KE at the bottom: mgh = ½mv², so v = √(2gh). A 5 m slide gives v = √(2 × 9.81 × 5) = 9.9 m/s, ignoring friction.
Where potential energy shows up in the real world
Hydroelectric power is the biggest commercial application of gravitational potential energy. Water in mountain reservoirs holds enormous PE; falling through turbines, it converts to rotational kinetic energy and then to electricity. Worldwide hydro generates about 4,200 TWh per year, 15 percent of global electricity.
- hydroelectric dam = water at height converts to electricity
- roller coaster = lift hill stores PE, released as the ride accelerates downhill
- archery bow = elastic PE in the limbs launches the arrow
- clock spring = wound spring stores energy to drive gears
- car suspension = springs absorb bumps then release the energy
- pole vault = athlete's kinetic energy bends the pole into elastic PE, then lifts them
- pumped-storage hydro = electricity used to pump water uphill, storing energy as PE
Pumped-storage hydro is essentially an energy battery. When electricity is cheap (overnight, or during high solar/wind output), water is pumped from a low reservoir to a high one. When demand spikes, the water flows back down through turbines. Round-trip efficiency is 70-85 percent, making PE storage the most widespread grid-scale energy storage technology by far.
Common potential energy mistakes
Three errors dominate physics homework. First, choosing an inconsistent reference height. If you set h = 0 at the top of a hill and then plug positive heights for everything below, your PE values come out wrong-signed. Either be careful with signs or define h = 0 below everything.
Second, forgetting friction. PE = KE only in a frictionless world. On a real slide some PE turns into heat and sound. The roller coaster never returns to its starting height on the other side of a valley because energy has leaked out.
PE = mgh treats gravity as constant. At low altitudes that is accurate within a few percent. For satellites, mountains, or any height comparable to Earth's radius (6,371 km), use the inverse-square form U = -GMm/r instead. The 9.81 m/s² figure underestimates the energy needed to reach orbit by a factor of two or more.
Third, using the wrong formula for elastic PE. The factor of ½ is essential — leaving it out doubles the answer. Likewise, x must be measured from the spring's natural (unstretched) length, not from its current position under load.